#511968

order	512
nilpotency class	4
isoclinism	isoclinic class of #511966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32930
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#32948	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3573 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

934 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 934 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(2,8)(5,21)(6,22)(9,23)(10,24)(11,25)(12,29)(13,28)(14,37)(15,40)(17,41)(18,43)(20,44)(26,27)(30,42)(31,45)(32,56)(33,48)(34,53)(35,51)(36,54)(38,55)(46,57)(47,49)(50,52)(58,63)(59,61)(62,64);(1,2,9,29,6,12,5,8,7,13,21,27,22,26,23,28)(3,10,30,53,17,34,15,24,16,33,40,49,41,47,42,48)(4,11,31,54,20,36,18,25,19,35,43,52,44,50,45,51)(14,32,55,64,39,58,38,46,37,57,61,63,60,62,59,56);(1,3)(2,10)(4,14)(5,15)(6,17)(7,16)(8,24)(9,30)(11,32)(12,34)(13,33)(18,38)(19,37)(20,39)(21,40)(22,41)(23,42)(25,46)(26,47)(27,49)(28,48)(29,53)(31,55)(35,57)(36,58)(43,61)(44,60)(45,59)(50,62)(51,56)(52,63)(54,64);(1,4)(2,11)(3,14)(5,18)(6,20)(7,19)(8,25)(9,31)(10,32)(12,36)(13,35)(15,38)(16,37)(17,39)(21,43)(22,44)(23,45)(24,46)(26,50)(27,52)(28,51)(29,54)(30,55)(33,57)(34,58)(40,61)(41,60)(42,59)(47,62)(48,56)(49,63)(53,64)

Polycyclic presentation:

2^9|b^2=ehi,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,h^a=hi,g^c=gi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72341930
maximal subgroup of	#72320061, #72341930, #72342004, …
maximal normal subgroup of	#72320061, #72341930, #72343955, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419164);